Keep the smooth

β = β_{H}

geometry fixed away from the horizon but identify Euclidean time with arbitrary period

β

. The horizon then has deficit angle

2 π (1 - β / β_{H})

. Its delta-function curvature contributes

\int_{M} g R_{tip} = 4 π (1 - \frac{β}{β _{H}}) A,

(1)

and hence

I_{tip} = - \frac{A}{4 G _{N}} (1 - \frac{β}{β _{H}}) .

(2)

Using

lo g Z = - I_{grav}

S_{BH} = (1 - β \partial_{β}) lo g Z ∣_{β_{H}} = \frac{A}{4 G _{N}} = \frac{2 π ^{2} r _{H}^{3}}{4 G _{N}} .

(3)

All smooth bulk terms, including the cosmological-constant volume term, are proportional to the Euclidean time period and are annihilated by

1 - β \partial_{β}

. The asymptotic Gibbons–Hawking–York boundary term and holographic counterterms are likewise smooth and linear in

β

. Only the curvature singularity at the fixed point of the Euclidean time circle survives.

Varying the boundary period and filling it by the corresponding smooth Euclidean saddle computes the same canonical partition function. Because each bulk metric obeys the Einstein equation, the implicit first-order metric variation of the on-shell action reduces to boundary terms; regularity relates the varied horizon radius to the varied period. The thermodynamic identity

S = βE - I_{E} = (1 - β \partial_{β}) lo g Z

(1)

then yields the same Bekenstein-Hawking entropy. The conical method is an off-shell way to isolate the local horizon term, while the smooth-saddle method packages that term into the variation of the entire solution.